Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication
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The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in Õ(n^7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n^7/4) bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time Õ(n^3/2) storing just Õ(n^3/2) bits on the side. We then complement our data structure with a lower bound showing that any data structure storing r bits on the side, with n < r < n^2 must have query time t satisfying t r = Ω̃(n^3). For r ≤ n, any data structure must have t = Ω̃(n^2). Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over F_2.
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